Dear David,

I was trying to solve the indefinite one. This is just an attempt to understand the working of Firedrake. We hope to solve PDEs on a sphere, for example the Shallow Water Equation on Icosahedral grids with time; and I am glad that such a geometry can be worked on using Firedrake.

David
On 10 Dec 2015, at 15:53, David Ham <David.Ham@imperial.ac.uk> wrote:

Dear David,

I think you have confused what Lawrence was asking. Since you are working with Sebastian Reich, we assume you are solving atmospheric problems. In this context, the definite Helmholtz equation usually occurs:

laplacian u - u = f

while you appear to be solving the indefinite Helmholtz equation:

laplacian u + u = f

We were just attempting to ascertain whether you were deliberately solving the indefinite version of the problem. Is this the case?

David

On Thu, 10 Dec 2015 at 14:47 Angwenyi David <kipkoej@gmail.com> wrote:
Integration by parts is equivalent to using Green’s function; and so it turns out as follows:

laplacian u = - u + f
Greens function: integ((laplacian u)v) + integ(dot(grad u, grad v)) = surfinteg((partial_n u)v)

So that the weak form is obtained by substituting laplacian u = -u + f from the Helmholtz equation in the Green’s function. The same is arrived at by integration by parts.

Sincerely,

David

> On 10 Dec 2015, at 15:23, Lawrence Mitchell <lawrence.mitchell@imperial.ac.uk> wrote:
>
>
>
> On 10/12/15 14:12, Angwenyi David wrote:
>>>>> nx = 10
>>>>> ny = 10
>>>>> nz = 20
>>>>> mesh = BoxMesh(nx, ny, nz, 1.0, 1.0, 2.0)
>>>>> V = FunctionSpace(mesh, "CG", 1)
>>>>> u = TrialFunction(V)
>>>>> v = TestFunction(V)
>>>>> f = Function(V)
>>>>> f.interpolate(Expression("(2*x[0])+(3*x[1])+(0.5*x[2])"))
>> Coefficient(FunctionSpace(Mesh(VectorElement('Lagrange', tetrahedron,
>> 1, dim=3), 3), FiniteElement('Lagrange', tetrahedron, 1)), 4)
>>>>> a = (dot(grad(v), grad(u)) - v * u) * dx
>
> This is the indefinite form of the Helmholtz equation.  Did you
> actually want the sign-positive Helmholtz equation?
>
> The later has strong form:
>
> -laplacian u + u = f
>
> When integrating by parts the weak form becomes:
>
> <grad u, grad v> + <u, v> = <f, v>
>
> As to incorporation of boundary conditions, see
>
> http://firedrakeproject.org/variational-problems.html#incorporating-boundary-conditions
>
> Cheers,
>
> Lawrence
>
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